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Exam 4 \hspace{40mm}Name: \underline{\hspace{30mm}}

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Directions:  Answer each question to the best of your ability.  The point value of each question follows the question.  Calculators are allowed, but you must show appropriate work to receive full credit.  

Choose the best response for the following 9 questions. (5 pts each)
\begin{enumerate}
\item In the series, $a+ar+ar^2+r^3+...$, for which values of $r$ will the sum have a finite value?

A. $\frac{2}{3}$  \hfill  B. 5  \hfill  C. $-\frac{4}{3}$  \hfill D. $-\frac{1}{5}$ \hfill  E.  1 
\begin{enumerate}
\item A, B, E
\item A
\item A, C
\item A, D
\end{enumerate}

\item Which of the following series are geometric?
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A. $1+\frac{5}{6}+\frac{2}{3}+\frac{1}{2}+...$ \hfill  B. $2-\frac{2}{3}+\frac{2}{9}-\frac{2}{27}+...$ \hfill C. $\frac{2}{7}+\frac{3}{14}+\frac{4}{28}+\frac{5}{56}...$ \hfill D.  $\frac{3}{4}+\frac{9}{20}+\frac{27}{100}+\frac{81}{500}...$
\end{small}

\begin{enumerate}
\item A, B
\item C, D
\item B, D
\item A, C
\end{enumerate}

\item When the pump goes online, oil is retrieved from a well at a rate of 10,000 $brl/day$.  However, each day of pumping reduces the volume retrieved by .2\% per day.  Which formula correctly finds the amount of oil retrieved at the end of 30 days? 
\begin{enumerate}
\item $\frac{10,000(1-.002^{30})}{.002}$
\item $\frac{10,000(1-.998^{30})}{.002}$
\item $\frac{10,000(1-.998^{30})}{.998}$
\item $\frac{10,000(1-.002^{30})}{.998}$
\end{enumerate}

\item What restriction exists for the Taylor polynomial to approximate a given function?
\begin{enumerate}
\item The function must be defined on the domain
\item The function must be positive-valued for all real numbers
\item The function must have a derivative that cycles as you take more derivatives
\item There is no restriction
\end{enumerate}

\newpage
\item What is the sum of the following geometric series?   $2-\frac{2}{5}+\frac{2}{25}-\frac{2}{125}+...$
\begin{enumerate}
\item $\frac{5}{6}$
\item $1$
\item $\frac{5}{3}$
\item $\frac{5}{2}$
\end{enumerate}

\item Accounting for a possible $0^{th}$ term, which is the formula for the $n$th term of the following series?   $1-x^2+\frac{x^4}{2!}-\frac{x^6}{3!}+...$
\begin{enumerate}
\item $\frac{(-1)^{(n-1)}x^{2n}}{(n-1)!}$
\item $\frac{(-1)^nx^{2n}}{n!}$
\item $\frac{(-1)^{(n-1)}x^{2(n-1)}}{(n-1)!}$
\item $\frac{(-1)^{(n-1)}x^{2(n-1)}}{n!}$
\end{enumerate}
\item Which is the correct Taylor series for $e^x$ about $x=0$
\begin{enumerate}
\item $1+\frac{x}{2!}+\frac{x^2}{3!}+\frac{x^3}{4!}+\frac{x^4}{5!}+...$
\item $1+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+...$
\item $x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$ 
\item $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$
\end{enumerate}
\item Which is the correct Taylor series for $\sin x$ about $x=0$
\begin{enumerate}
\item $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$
\item $x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...$ 
\item $1-\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+...$ 
\item $1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\frac{x^4}{4!}+...$
\end{enumerate}
\item Which is the defining characteristic of a Taylor polynomial of degree $n$ that approximates $f(x)$ about $x=a$?
\begin{enumerate}
\item The coefficient for each term of the polynomial is nonzero.
\item The polynomial is an even function.
\item The polynomial has $n$ real roots.
\item The evaluation of the $1^{st}$ to $n^{th}$ derivatives at $x=a$ of the polynomial and $f(x)$ are the same.
\end{enumerate}
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\item State the formula for the degree 3 Taylor polynomial of $f(x)$ about $x=0$  (5 pts)
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\item Find the first 4 nonzero terms of the Taylor series about $x=0$ for the following functions (5 pts each)
\begin{enumerate}
\item $\cos x$
\vfill
\item $ \frac{1}{(1+x)^3}$

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\end{enumerate}
\item Use your answer from 11a to estimate $\cos 1$ to 6 decimal places.  (5 pts)
\vspace{20mm}
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\item Use the Taylor polynomial of degree 4 about $x=0$ for $e^x$ to estimate $e^1$ to 6 decimal places.  (5 pts)
\vspace{20mm}
\item Calculate the error on the degree 4 Taylor polynomial estimate of $e^1$ to 6 decimal places.  (5 pts)
\vspace{15mm}
\item Find the necessary degree for a Taylor polynomial about $x=0$ to calculate $e^1$ within .000001. (5 pts)
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\item Using your answer to 15, estimate $e^1$ to 8 decimal places.  (5 pts)
\vspace{20mm}
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For the last part of the test, choose between pages 5 and 6.  I will grade only one.  If you do both, indicate which you want me to grade.  For each of the applicable problems, use the first 4 non-zero terms for the series.
\item Express $f(x)=e^{x^3}$ as a Taylor series.  (3 pts)
\vspace{20mm}
\item Verify using Taylor series that $f'(x)=3x^2e^{x^3}$  (7 pts)
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\item Give the formula for the $n$th term for the Taylor series for $3x^2e^{x^3}\,dx$.  (5 pts)
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\newpage

\item Find the Taylor series for $\frac{1}{\sqrt{1+y}}$.  (5 pts)
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\item Use your answer to \#20 to find the Taylor series for $\frac{1}{\sqrt{1-x^2}}$.  (5 pts)
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\item Use your answer to \#21 to find the Taylor series for $\arcsin x$.  (5 pts)
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\end{enumerate}
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